Every group admits an initial homomorphism to a pi-powered group
Suppose is a group and is a set of primes. There exists a -powered group and a homomorphism of groups such that for any -powered group and any homomorphism , there is a unique homomorphism such that .
We can think of the functor that sends to the group as the free -powering functor. It is the left-adjoint functor to the forgetful functor from -powered groups to groups.
In the localization terminology
The term -local is sometimes used to refer to a group that is powered over all the primes not in . The functor described above is, in that context, termed the -localization functor. By we mean the complement of in the set of all primes.
- The free powered group for a set of primes can be thought of as being obtained from the abstract free group by applying the free -powering functor.
We are interested in cases where the canonical homomorphism from the group to its -powering is injective, and in related questions, some of which are explored below.
|Group assumption on both the starting group and the big group||Divisibility/powering/torsion assumption on the starting group||Divisibility/powering/torsion assumption on the big group||Is this always possible?||Proof|
|nilpotent group||none||divisible group||No||nilpotent group need not be embeddable in a divisible nilpotent group|
|nilpotent group||-torsion-free group (equivalently, -powering-injective group)||-powered group||Yes||every pi-torsion-free nilpotent group can be embedded in a unique minimal pi-powered nilpotent group|
|arbitrary group||none||divisible group||Yes||every group is a subgroup of a divisible group, every pi-group is a subgroup of a divisible pi-group|
|arbitrary group||-torsion-free group (equivalently, -powering-injective group)||-powered group||No||Powering-injective group need not be embeddable in a rationally powered group|